In many applications in the manipulation of charged particle beams, the separation of the constituents of the beam by their mass, energy or charge is required. Magnet separators or sectors are often used to achieve this. Such magnet separators are used in mass and energy spectrometers. These magnet separators employ uniform fields perpendicular to the incident charged particle. Those skilled in the art of magnetic design go to great lengths to ensure uniformity. Charged particles in a uniform field follow curved trajectories. The trajectory that a charged particle with a mass m, an energy E, and a net charge q follows is given by the following equation:                               R          2                =                  C          ⁢                      1                          B              2                                ⁢                      (                          mE              q                        )                                              (        1        )            
Where R is the radius of the trajectory of the charged particle, or radius of curvature of the charged particle, and C is a constant of proportionality dependent upon the units of the parameters. The dependence of the square of the radius of curvature R in Equation (1) upon the mass-energy-to-charge ratio mE/q results in a dispersion of the charged particles entering into a uniform field according to the square root of their various mE/q ratios.
Depending upon the specific charged-particle separation application, many adaptations and embodiments of the uniform field magnet separator are employed. Mass spectrometers, for example, may use uniform magnet separators with permanent magnets or electromagnets to achieve a spatial separation of ions according to their mass and charge when accelerated to a fixed energy. The advantage of the uniform magnetic separator is that for a collimated charged particle beam it provides a focus along a plane parallel to the magnetic field along which the particles of all mE/q are focused. This plane lies at an angle of 45 degrees from the initial input beam trajectory. That is to say, that the trajectories of parallel charged particles of equivalent mE/q converge after the particles have followed an arc of their trajectories of 135 degrees from initial contact with the magnetic field. The disadvantage of the uniform field magnetic sector is that the separation of adjacent particles with mE/q differing by fixed amounts is a non-linear function of position. That is to say, larger mass-energy-to-charge ratios lie significantly closer than lower ratios.
For non-collimated charged particle beams, the uniform-field magnet sector is often modified to include a transverse gradient which provides focusing to compensate for the non-collimation.